By making the keys "naturally comparable"
Supposing that we have a key function that implements the natural-sort comparison, as in Claudiu's answer for the related question:
import re
def natural_sort_key(s, _nsre=re.compile('([0-9]+)')):
    return [int(text) if text.isdigit() else text.lower()
            for text in _nsre.split(s)]
Then we can create a wrapper class for strings which is compared using that function, transform the keys of the dict, and proceed as before:
from functools import total_ordering
@total_ordering
class NaturalSortingStr(str):
    def __lt__(self, other):
        return natural_sort_key(self) < natural_sort_key(other)
fixed = {NaturalSortingStr(k):v for k, v in data.items()}
print(json.dumps(fixed,sort_keys=True,indent=2))
Note that functools.total_ordering is introduced in Python 3.2. In older versions, we should instead define __gt__, __le__ and __ge__ explicitly, in corresponding ways. (Python's sort algorithm should not use these, but it is a good idea to include consistent definitions for correctness.) Of course, the base str's implementations of __eq__ and __ne__ do not need to be replaced.
(In 2.7 we could also instead implement a corresponding __cmp__, but this will break in 3.x.)
By putting the keys in order first
In 3.7 and up, dictionary keys are guaranteed to preserve their order; in earlier versions, we can use collections.OrderedDict to get that property. Note that this does not sort keys, but maintains the order of insertion.
Thus, we can determine the necessary order for keys, and create a new dict by inserting keys in that order:
import sys
if sys.version_info < (3, 7):
    # In newer versions this would also work, but is unnecessary
    from collections import OrderedDict as odict
else:
    odict = dict
sorted_keys = sorted(data.keys(), key=natural_sort_key)
sorted_data = odict((k, data[k]) for k in sorted_keys)
print(json.dumps(sorted_data, indent=2))
Since the data was sorted ahead of time, sort_keys=True is no longer necessary. In modern versions, since we are using the built-in dict, we could also write a dict comprehension (rather than passing a generator to the constructor).